On $k_\mathbb{R}$-spaces and $s_\mathbb{R}$-spaces
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866912437715861504 |
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| author | Gabriyelyan, Saak Reznichenko, Evgenii |
| author_facet | Gabriyelyan, Saak Reznichenko, Evgenii |
| contents | We give new characterizations of spaces $X$ which are $k_\mathbb{R}$-spaces or $s_\mathbb{R}$-spaces. Applying the obtained results we provide some sufficient and necessary conditions on $X$ for which $C_p(X)$ is a $k_\mathbb{R}$-space or an $s_\mathbb{R}$-space. It is proved that $C_p(X)$ is a $k_\mathbb{R}$-space for any space $X$ with one non-isolated point; if, in addition, $|X|$ is not sequential, then $C_p(X)$ is even an $s_\mathbb{R}$-space. Under $(CH)$, it is shown that there exists a separable metrizable space $X$ such that $C_p(X)$ is an Ascoli space but not a $k_\mathbb{R}$-space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_15206 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On $k_\mathbb{R}$-spaces and $s_\mathbb{R}$-spaces Gabriyelyan, Saak Reznichenko, Evgenii General Topology 54A05, 54B05, 54C35, 54D50 We give new characterizations of spaces $X$ which are $k_\mathbb{R}$-spaces or $s_\mathbb{R}$-spaces. Applying the obtained results we provide some sufficient and necessary conditions on $X$ for which $C_p(X)$ is a $k_\mathbb{R}$-space or an $s_\mathbb{R}$-space. It is proved that $C_p(X)$ is a $k_\mathbb{R}$-space for any space $X$ with one non-isolated point; if, in addition, $|X|$ is not sequential, then $C_p(X)$ is even an $s_\mathbb{R}$-space. Under $(CH)$, it is shown that there exists a separable metrizable space $X$ such that $C_p(X)$ is an Ascoli space but not a $k_\mathbb{R}$-space. |
| title | On $k_\mathbb{R}$-spaces and $s_\mathbb{R}$-spaces |
| topic | General Topology 54A05, 54B05, 54C35, 54D50 |
| url | https://arxiv.org/abs/2506.15206 |